(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

For the differential equation, verify (by differentiation and substitution) that the given function y(t) is a solution.

2. Relevant equations

[itex] y' - 4ty = 1 [/itex]

[itex] y(t) = \int_{0}^{t} e^{-2(s^{2}-t^{2})} ds[/itex]

3. The attempt at a solution

I attempted to take [itex]\frac{d}{dt}[/itex] of y(t) as usual but

1. if I do not try bringing the [itex]\frac{d}{dt}[/itex] inside the integral I can do nothing because there is no elementary antiderivative of y(t).

2. if I do bring the [itex]\frac{d}{dt}[/itex] inside the integral, I can use the chain rule to get

[itex] y(t) = (-2) \int_{0}^{t} (-2t) e^{-2(s^{2}-t^{2})} ds[/itex]

but since my variable of integration is ds not dt, this doesn't allow me to use a u-substitution as I had hoped nor can I think of a way to relate ds and dt.

More or less I do not know how to take [itex]\frac{d}{dt}[/itex] of y(t) and I do not know any other ways to solve the problem.

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# Veryfing ODE for complicated y(t)

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